# Authentic Problems: Expanded Square

ACMNA077; ACMMG087; ACMMG088; ACMMG091

This unit for Year 4 is one of a set of ten units in the special topic “Mathematical Inquiry into Authentic Problems”. Each of these units is designed around the 4D Guided Inquiry Model, and highlights the importance of students providing mathematical evidence. The lessons adopt a carefully designed pedagogy to help students master content knowledge whilst learning about the process of inquiry.

Expanding a square is an art technique that flips cut-out shapes from the inside of a coloured paper square to the outside. Students must design an expanded square that flips out about half the area. They explore different methods to determine whether they have removed about half of the original square and adjust their design accordingly. When sharing their designs they describe any symmetry created and the transformations used and justify their area calculations.

### Lesson 1: Discover

Students are introduced to expanded squares (based on the Eastern art of Notan) and explore their characteristics. They construct their first expanded square and discuss its features, including symmetry.

### Lesson 2: Devise

Students use fractions to describe the amount of white space in some partially coloured grid squares, and then in some expanded square examples. They consider what fraction of white space is best for visual balance. Students begin to design an expanded square that has approximately half the original square flipped to the outside.

### Lesson 3: Develop

Students produce mathematical evidence to convince others that approximately half the area of their original square has been flipped to the outside. They find areas by counting grid squares, or covering with 1 cm cubes, and perhaps rearranging shapes. They seek constructive feedback on the method they used to determine the fraction of the area that has been flipped and on the mathematical evidence they recorded.

### Lesson 4: Defend

Students adjust their designs (or describe possible adjustments) and ensure their evidence is detailed and organised clearly so it can be displayed for others to critique. During the feedback session students provide constructive feedback on the strengths and weaknesses of the mathematical evidence and designs. They reflect on feedback given on their display and on their learning throughout the inquiry.

Last updated May 22 2018.